Problem: Divide the following complex numbers. $ \dfrac{-10+30i}{-5+5i}$
Answer: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-5-5i}$ $ \dfrac{-10+30i}{-5+5i} = \dfrac{-10+30i}{-5+5i} \cdot \dfrac{{-5-5i}}{{-5-5i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(-10+30i) \cdot (-5-5i)} {(-5+5i) \cdot (-5-5i)} = \dfrac{(-10+30i) \cdot (-5-5i)} {(-5)^2 - (5i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(-10+30i) \cdot (-5-5i)} {(-5)^2 - (5i)^2} = $ $ \dfrac{(-10+30i) \cdot (-5-5i)} {25 + 25} = $ $ \dfrac{(-10+30i) \cdot (-5-5i)} {50} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({-10+30i}) \cdot ({-5-5i})} {50} = $ $ \dfrac{{-10} \cdot {(-5)} + {30} \cdot {(-5) i} + {-10} \cdot {-5 i} + {30} \cdot {-5 i^2}} {50} $ Evaluate each product of two numbers. $ \dfrac{50 - 150i + 50i - 150 i^2} {50} $ Finally, simplify the fraction. $ \dfrac{50 - 150i + 50i + 150} {50} = \dfrac{200 - 100i} {50} = 4-2i $